Problem 15 - University Calculus: Early Transcendentals

Question

Sketch the surfaces CYLINDERS $$x^{2}+4 z^{2}=16$$

Step-by-Step Solution

Answer

The surface is an elliptical cylinder oriented along the y-axis.

Step 1: Rewrite the equation

First, let's take a look at the given equation: \[x^2 + 4z^2 = 16\]. We need to rewrite it in a familiar form by dividing each term by 16: \[\frac{x^2}{16} + \frac{z^2}{4} = 1\]. This equation is now in the form of an ellipse in the \(xz\)-plane.

Step 2: Identify the shape of the cross-section

Notice that the equation \(\frac{x^2}{16} + \frac{z^2}{4} = 1\) describes an ellipse. In the \(xz\)-plane, the semi-major axis parallel to the \(x\)-axis is 4, and the semi-minor axis parallel to the \(z\)-axis is 2.

Step 3: Extend along the y-axis

Since there is no \(y\) term in the equation, the elliptical cross-section described is constant for all real \(y\) values. This means that the surface is a cylinder extending infinitely along the \(y\)-axis.

Step 4: Sketch the surface

To sketch the surface: 1. Draw the ellipse in the \(xz\)-plane with center at (0,0), semi-major axis 4 along x, and semi-minor axis 2 along z. 2. Extend this ellipse along the \(y\)-axis to form a cylindrical shape. The result is an elliptical cylinder with its axis along the \(y\)-axis.

Key Concepts

Sketching Surfaces

When sketching surfaces in calculus, it's important to first understand the type of surface you are dealing with. For the exercise, we start by analyzing the given equation, \(x^2 + 4z^2 = 16\). This equation represents a surface in the three-dimensional coordinate system that requires careful plotting.

The key to sketching surfaces is transforming the equation into a more recognizable form. By dividing the entire equation by 16, we convert it into \(\frac{x^2}{16} + \frac{z^2}{4} = 1\). This step helps in identifying the correct surface to sketch.

Once transformed, you can determine the features of the surface, such as where it will intersect the axes and how it spreads in space. In this example, it becomes evident that the surface forms an ellipse along the \(xz\)-plane, which then extends infinitely along the \(y\)-axis, forming an elliptical cylinder. Keeping these techniques in mind will make sketching complex surfaces much more approachable.

Elliptical Cylinders

Elliptical cylinders are a fascinating type of surface in calculus. They are defined by their elliptical cross-sections, and have some unique characteristics. The key feature of an elliptical cylinder is that it has a constant cross-section along one of its axes, in this case, the \(y\)-axis.

Let's consider our transformed equation, \(\frac{x^2}{16} + \frac{z^2}{4} = 1\). This equation describes an ellipse in the \(xz\)-plane. To fully grasp the shape of this cylinder, it's crucial to understand the components of the ellipse formed by this equation:

• The semi-major axis along the \(x\)-axis is 4. This is computed from the square root of 16, which is under the \(x^2\) term in the denominator.
• The semi-minor axis along the \(z\)-axis is 2, derived from the square root of 4 under the \(z^2\) term.

An elliptical cylinder like the one in this problem stretches infinitely along the \(y\)-axis, as there is no \(y\) term involved in the equation. The ellipse remains constant along every point of the \(y\)-direction, showing how an elliptical cylinder can be effectively visualized.

Cross-Section Analysis

Cross-section analysis is a powerful tool in understanding and sketching surfaces described by mathematical equations. For this specific problem, the given equation was converted into an elliptical equation, \(\frac{x^2}{16} + \frac{z^2}{4} = 1\), to analyze it in the \(xz\)-plane.

In cross-section analysis, we examine slices—or cross-sections—of the surface parallel to certain coordinate planes. The cross-section provides insight into the dimensions and shape of the surface at any point along an axis. Here are some essential takeaways for the ellipse in this context:

• In the \(xz\)-plane, this cross-section is elliptical, revealing its semi-major and semi-minor axes.
• In absence of a \(y\) term, these elliptical cross-sections span infinitely along the \(y\)-axis, demonstrating the cylindrical nature of the surface.

By utilizing cross-section analysis, you can more easily sketch and understand the full nature of the three-dimensional surface. It's a skill that enhances visualization and interpretation of surfaces in calculus and beyond.